## Monday, June 16, 2008

### The other Uncertainty Principle

Update:  I have to say that out of all my science posts, this is the one I regret the most.  I'm not sure that I would say it's inaccurate, but I think it is incomplete.  A more complete explanation would have explained the uncertainty principle in a way that makes sense with virtual particles.  But I will not attempt to fix the post; it remains below unchanged.

I've previously discussed the Uncertainty Principle:

Δx Δp ≥ ħ/2

Δx is the uncertainty in the position of a particle, while Δp is the uncertainty of the momentum of the particle. ħ/2 is a very small fundamental constant.

There is another Uncertainty Principle that people are fond of mentioning.

ΔE Δt ≥ ħ/2

Here, E is energy and t is time. This equation is often mentioned in footnotes, where writers note that not only is space uncertain, but time is also uncertain. The general thrust is "Quantum Mechanics: isn't it insane? Let us all marvel at Nature's imagination!"

The problem is that the equation is so often misunderstood. The equation is not exactly a myth, but it does not mean what people think it means.

Position (x), momentum (p) and energy (E) are all measurable quantities. Time (t) is not. What does it mean to measure the "time" of a particle? Ok, it is possible to measure "time" if you use the Theory of Relativity, but this equation has nothing to do with that. It is based on non-relativistic equations, and works for particles moving much slower than the speed of light. Because time is not a measurable quantity, the second uncertainty principle means something entirely different from the first one (and has a different derivation too).

Δt does not mean the same as Δx. Δx is defined as the average difference between the measured value of x and the expected value of x. Δt is defined as the amount of time it takes for a wavefunction to change by a "significant"* amount. If the uncertainty in energy (ΔE) is very low, then it will take a long time for the wavefunction to change. If the wavefunction changes very quickly (Δt is very small), then the energy must be uncertain. If the energy is known exactly, then the wavefunction does not change at all in any measurable way. This is called a "stationary state": a wavefunction that does not move, because the energy is exactly known.

In some sense, Δt really does indicate some sort of uncertainty. If you have a moving particle, you can use its position to determine how much time passes. Any such "clock" would have an uncertainty in time equal to Δt. This isn't due to any uncertainty in time, but rather, the uncertainty in position. If you have a particle in a stationary state, you can't use it as a clock at all, because its position never changes, and Δt is infinite. But is very misleading to think Δt means the same thing as uncertainty in time. Time itself is not uncertain--that's just your clock.

While Quantum Mechanics is weird, it's not quite so weird that it denies universal time. No, it was Special Relativity that denied universal time. Glad that's settled, then...

*"Significant" change means that a measurable quantity changes by one standard deviation. For example Δt could mean the time required for the expected position (x) to change by a quantity Δx.